separability is required for integral closures to be finitely generated
Let where the are indeterminates.
Let be the subring of (power series in over ) consisting of
such that generates a finite extension of . Then every element of is a power of times a unit (first factor out the largest power of . What’s left is ; its inverse is ). Hence the ideals of are powers of , so is a PID (in fact, it is a DVR).
Now, let . Now, because it uses all of the and thus the coefficients do not define a finite extension of . However, is integral over : which implies that the degree of over the field of fractions is . Hence a basis for is . There are other elements integral over :
|Title||separability is required for integral closures to be finitely generated|
|Date of creation||2013-03-22 17:02:15|
|Last modified on||2013-03-22 17:02:15|
|Last modified by||rm50 (10146)|